• Density of Primes in 𝑙-th Power Residues

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/123/01/0019-0025

• # Keywords

Distribution of primes; higher residue symbols.

• # Abstract

Given a prime number 𝑙, a finite set of integers $S=\{a_1,\ldots,a_m\}$ and 𝑚 many 𝑙-th roots of unity $\zeta^{r_i}_l,i=1,\ldots,m$ we study the distribution of primes 𝑝 in $\mathbb{Q}(\zeta_l)$ such that the 𝑙-th residue symbol of $a_i$ with respect to 𝑝 is $\zeta^{r_i}_l$, for all 𝑖. We find out that this is related to the degree of the extension $\mathbb{Q}\left(a^{\frac{1}{l}}_1,\ldots,a^{\frac{1}{l}}_m\right)/\mathbb{Q}$. We give an algorithm to compute this degree. Also we relate this degree to rank of a matrix obtained from $S=\{a_1,\ldots,a_m\}$. This latter argument enables one to describe the degree $\mathbb{Q}\left(a^{\frac{1}{l}}_1,\ldots,a^{\frac{1}{l}}_m\right)/\mathbb{Q}$ in much simpler terms.

• # Author Affiliations

1. The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113, India

• # Proceedings – Mathematical Sciences

Volume 132, 2022
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019